Concrete is the obvious choice. Ali F. Rhuzkan, a New York-based structural engineer, estimated in an article for National Memo that a 1,900-mile wall - seemingly Mr Trump's original plan - would require about 339 million cubic feet (12.5 million cubic yards) of concrete - three times more than the Hoover Dam.

Mr Rhuzkan's estimate was based on a wall that ran five feet beneath the ground and 20 feet above. Mr Trump's claims for the wall range between 30 feet and, more recently, 55 feet. Even at 1,000 miles - vast amounts of concrete would be required.

That concrete, Mr Rhuzkan wrote, would need to be manufactured in slabs nearby - likely in dedicated plants - and transported to building sites, with all the enormous production and staffing costs involved.

What are the costs?

"I would build a great wall, and nobody builds walls better than me, believe me, and I build them very inexpensively."

Mr Trump claims the total cost of the wall will be $10 (£7.5) billion to $12 billion. But estimates from fact checkers and engineers seem to be universally higher.

The 650 miles of fencing already put up has cost the government more than $7 billion, and none of it could be described, even charitably, as impenetrable, physical, tall, powerful, or beautiful.

There are other reasons the costs would be likely to escalate: Mr Trump's plans require extending the wall into increasingly remote and mountainous regions, raising the building costs substantially.

Adding even more to the expense, the new 1,000 miles would crisscross private land, which would have to be purchased, perhaps by legal force, or financial settlements made with owners.

A study by the Washington Post estimated the cost of Mr Trump's wall would be closer to $25 billion.

Former Mexican president Vicente Fox said: "I'm not going to pay for that ... wall," using a somewhat unpresidential word to make his feelings on the matter clear

Asked by US journalists how he could force Mexico to pay, Mr Trump suggested he could ransom the country by blocking undocumented immigrants from sending money home, using a provision of the US Patriot Act designed to stop funding for terrorism.

Pressed on whether he could really use anti-terror legislation to ransom Mexico into paying for a wall, Mr Trump has variously suggested other options including increasing the fees on visa applications, charging more for border crossing cards and enforcing trade tariffs.

Asked whether any of his solutions were realistic, he told the Washington Post: "It's realistic if you know something about the art of negotiating. If you have a bunch of clowns negotiating, it's not realistic."

International relations experts have questioned whether the security gain of building a huge wall along the border outweigh the security risk of alienating your southern neighbour.

What about the environment?

Free movement between the US and Mexico is not just a human issue. What would the construction of a wall mean for animals that live near the border?

The US-Mexico border region is a delicate ecosystem with regular animal and bird migrations moving between the north and south of the American continent.

A number of species need to cross the border to mate with their genetically different cousins, including the endangered North American jaguar and black bears, which would be threatened without being able to mate with Mexican bears.

The wall would also have to take into account natural flooding zones as well as large areas of sand, where the ground effectively moves.

Then there is the detrimental impact to the landscape of a massive construction project - digging, road building, and the appearance of a concrete wall up to 50 feet high, notwithstanding Mr Trump's pledge that it will be "very beautiful".